International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 08:23:58pm CET
Z10: flow problems
1:50pm - 2:10pm
Gradient jump penalty stabilisation of spectral/hp element method for under-resolved simulations
1Technological Institute of Aeronautics, Brazil; 2Imperial College London, United Kingdom; 3University College London, United Kindgon
One of the strengths of the discontinuous Galerkin (DG) formulation based on spectral/hp element spatial discretisations has been a balance of accuracy and robustness. The robustness arises out of their intrinsic property of high frequency diffusive behaviour of under resolved scales which is particularly useful in under-resolved flow simulations that naturally arise in high Reynolds number flows simulations. Similar properties are achievable through the addition of artificial diffusion, such as the spectral vanishing viscosity, for continuous Galerkin discretisations however at intermediate polynomials orders this approach is observed to be sub-optimal when compared to DG methods. In this paper we explore an alternative stabilisation approach of the introduction of an gradient jump penalisation (GJP) at the elemental boundaries. Analogous to DG methods this introduced a penalisation at the elemental interfaces as opposed to the interior element stabilisation of the SVV approach. Eigen-analysis of the GJP approach demonstrates that it has equivalent properties to DG-techniques which are superior to our previous SVV approaches. The favourable properties are also supported by two and three-dimensional simulations of linear advection problems and the turbulent incompressible Navier-Stokes simulations.
2:10pm - 2:30pm
Pressure-density compatible Riemann boundary condition for DG compressible flow simulations
1Imperial College London, United Kingdom; 2Beijing Aircraft Technology Research Institute of COMAC, Beijing, 102211, China
This analysis is motivated by designing better simulation tools to predict boundary layer transition at realistic Reynolds numbers over wings, which include surface irregularities such as steps, gaps, and bumps. To reduce the computational cost of high fidelity simulation of boundary layer flow, we wish to use a near body, reduced domain, whose outer boundary conditions are taken from a computationally cheaper 3D RANS simulation. For a subsonic inflow condition in a Discontinuous Galerkin (DG) and Riemann-based solver, two conditions can be imposed in the normal direction to the boundary. As the standard condition in a DG solver, the two incoming Riemann invariants are set, leading to a non-reflecting boundary. However, this does not adequately enforce a compatible pressure condition with the outer RANS simulation. This incompatibility in pressure distribution is undesirable since the pressure load is usually well captured by the Euler or RANS simulation (at least for lift prediction) and since the pressure distribution does not typically vary much over the boundary layer. The pressure distribution is therefore considered a reliable quantity of interest from the lower fidelity models and is an important property to maintain in the reduced domain. In this paper we therefore revisit the Riemann problem to see how best to enforce pressure and density compatibility at the inflow boundary and compare this condition with other choices of inflow condition. For validation we consider a reduced domain of a wing section normal to the leading-edge of the CRM-NLF model taken out of a full 3D RANS simulation at Mach 0.86 and a Reynolds number of 8.5 million. The results show that the new inflow condition leads to a better agreement not only on pressure distribution but also on velocity fields. On the other hand, using the standard Riemann-based inflow condition fails to give continuous velocity with the outer RANS fields.
2:30pm - 2:50pm
Spectral Element Investigation of Slat Cove Dynamics of Flow Past a 30P30N Three-Element High Lift Wing
1University of Ottawa; 2Wayne State University
A three-dimensional numerical analysis of low Reynolds number flow (O(10^4)) past a 30P30N high lift wing is carried out using a high order spectral element method. Such low Reynolds number flows have a significant impact on the design and development of small aerial vehicles but also increase our understanding of the transition from two-dimensional to three-dimensional structures in the flow, and from laminar to turbulent flow. We present this work as a demonstration of the advancement of high order methods in the pursuit of Direct Numerical Simulation (DNS) capabilities for complex geometries, such as this high lift aircraft wing configuration.
The current study is mainly focused on the flow in the slat cove region ahead of the main wing and the interaction of the slat wake and other shear layers. The vortical structures described in this study faithfully reproduce those observed in experiments. Görtler vortices are observed in the slat cove region for flows below a critical Reynolds number range as observed in experiments. For flows above the critical Reynolds number range, a roll-up is spotted in the slat cove region, and the existence of both streamwise and spanwise vortices is confirmed in the slat wake. Before the formation of Görtler vortices, three-dimensional vortical structures, both tongue- and rib-like depending on the Reynolds number, are observed in the slat cove region: these are similar to those encountered in the wake of flow past circular and square cylinders. These three-dimensional vortical structures are deemed to be responsible for the transition to three-dimensional flow. Streaks and spanwise vortices are observed in the slat wake above the critical Reynolds number range. These lead to the formation of hairpin vortices, which ultimately contribute to the transition to turbulent flow. Drag, lift, and pressure coefficient are studied, and relevant validation is carried out using available experimental data.
2:50pm - 3:10pm
On Hybridized Flux Reconstruction Schemes
Concordia University, Canada
Advances in high-order methods have enabled accurate and efficient numerical simulations of complex phenomena. Among these is the Flux Reconstruction (FR) approach, a family of schemes that depends on the choice of correction functions and may recover existing methods such as the Discontinuous Galerkin method for linear problems. FR methods are applied to conservation laws in divergence form. They are generally used in conjunction with explicit time-stepping for the temporal derivative, which, for stiff problems, demand very small time step sizes due to stability constraints. Hence, for this type of problem, implicit time-stepping is generally preferred. However, FR methods involve a large number of global degrees of freedom, which typically require solutions of prohibitively large nonlinear systems at every time step.
In this work, we present the hybridization of the FR approach for the Euler equations. The procedure consists of rewriting the initial formulation into local problems with a new variable added to the trace of the mesh, which approximates the conserved variable. In addition, the problem is augmented with a global conservation statement, which also provides a definition for the newly added trace variable. Hence, the problem is rewritten and reduced via static condensation, leading to a smaller system whose size depends only on the number of trace points.
We discuss two hybridization approaches, which we call Hybridized Flux Reconstruction (HFR) and Embedded Flux Reconstruction (EFR), in line with existing hybridized DG implementations. In both schemes, we consider discontinuous trace polynomials at the boundaries edges, further reducing the size of the system. We show that these two methods exhibit the expected orders of accuracy and significantly reduce the cost of implicit discretizations.
3:10pm - 3:30pm
Optimized Filters for Stabilizing High-Order Large Eddy Simulation
Concordia University, Canada
As stated in the Computational Fluid Dynamics (CFD) Vision 2030 report, published by the National Aeronautics and Space Administration (NASA), scale-resolving simulations are expected to play a vital role in the design of the next-generation aircraft. Current industry-standard CFD tools are predominantly based on finite-volume (FV) methods with second-order accuracy in space. In general, the FV method is often used due to its mature stabilization techniques for under-resolved and discontinuous flow features; however, it is dominated by point-wise indirect memory access, making FV memory bandwidth bound. High-order methods have been proposed as a solution to the relative inability of FV methods to harness modern hardware accelerators, although these schemes are typically less robust than lower-order FV schemes. High-order flux reconstruction (FR) schemes can be used to simulate unsteady turbulent flows using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) in the vicinity of complex geometries, and have been shown to be more suitable for modern high-performance computing hardware. However, the application of FR is often limited by non-linear instabilities, which can arise from the oscillatory behaviour of the underlying polynomial representation of the solution. These non-linear instabilities can be dealt with using a variety of techniques. In this study, we explore filtering and its parametrization for stabilizing under-resolved simulations of the Navier-Stokes equations. A new exponential filtering operator is proposed, which is normalized by the time-step size and designed to filter high-frequency modes. Filtering benefits from having a negligible additional computational cost; however, inappropriate choice of filtering parameters can result in accuracy degradation. In this study, we perform more than 14,000 numerical tests to obtain optimal sets of filtering parameters in the limit of infinite Reynolds number, with the objective being to stabilize simulations while maintaining high-order accuracy. We then verify that these optimal filters maintain high-order accuracy of the FR approach for non-linear problems using a series of simulations of an isentropic vortex, which is commonly used to verify the accuracy of a flow solver. The proposed optimized modally-filtered LES and non-filtered implicit LES of the isentropic vortex are compared. The computational domain is a cube of length 20, the center of the vortex is initialized at the coordinate origin, and the periodic boundary condition is used in all directions. The results show that the optimal filtering operators do not significantly degrade solution accuracy. Furthermore, the order of accuracy is maintained, and even more accurate solution is obtained using the filtering operator. Then, we compare filtered LES and non-filtered implicit LES simulations of the Taylor-Green vortex at the Reynolds number of 1600 and the Mach number of 0.1. The computational domain has a periodic boundary condition with nominally 64 degrees of freedom in each direction. We observe that the overall accuracy at each polynomial degree and for all filtering strengths is not impacted significantly. To explore the stabilization properties of the proposed filters and their accuracy for a set of benchmark problems, a previously unstable fully-developed turbulent channel flow is studied as validation for wall-bounded turbulent flows. There is a no-slip boundary condition applied to the walls and periodic boundaries in both the stream- and span-wise directions. The initial conditions are the Mach number of 0.3, bulk velocity Reynolds number of 6800 based on the channel half-width, and the friction velocity Reynolds number converges to 395. The total number of degrees of freedom for this simulation is 3/8 of that required for the DNS. We find that the filtering operators were suitable for LES of turbulent channel flow, and produced results consistent with the reference DNS data. This demonstrates that the filtered solutions can be even more accurate than unfiltered ones, while still stabilizing previously unstable simulations. Finally, we demonstrate the utility of these filters for more complex flows, specifically a stalled NACA 0020 airfoil. The flow over the airfoil at an angle of attack of 20 degrees, a Mach number of 0.2, and a Reynolds number of 20000 is studied using the modally-filtered LES and implicit LES. There are 68590 hexahedral elements and the domain has a periodic span of 0.45 of the chord length, which is sufficient for span-wise decorrelation. The modally-filtered LES results are in excellent agreement with the non-filtered implicit LES and reference DNS data. Based on these observations, we conclude proposing a set of optimal filtering parameters for performing LES simulations using the high-order FR approach. The proposed filtering operator is novel in the sense that its strength is made independent of the time-step size. Moreover, all of the filtering operators are optimized to yield the minimum filter strength for stabilization and preserve the order of accuracy.
3:30pm - 3:50pm
SEM's linear mechanism of energy transfer in Fourier space: insights into dispersion analysis for implicit LES
1Instituto Tecnológico de Aeronáutica, Brazil; 2National University of Singapore; 3Imperial College London
In recent years, different dispersion-diffusion (eigen)analyses have been developed and used to assess various spectral element methods (SEMs) with regards to accuracy and stability, which are especially important for under-resolved computations of transitional and turbulent flows. Not surprisingly, eigenanalysis has been used recurrently to probe the inner-workings of SEM-based implicit LES approaches, where numerical diffusion acts alone in small-scale regularisation. In this study we present and discuss an intriguing linear mechanism that causes energy transfer across Fourier modes as seen in the energy spectrum of SEM computations. Despite its linear nature, this mechanism has not been considered in eigenanalysis so far, possibly due to its connection to the often overlooked multiple-eigenvalues feature of temporal eigenanalyses. As we unveil the mechanism in the simplified context of linear advection, we point out how its effects might take place in actual turbulence simulations. In particular, we highlight how taking it into account in eigenanalysis should significantly improve dissipation estimates in wavenumber space, potentially allowing for a superior correlation between dissipation profiles and energy spectra measured in SEM-based under-resolved turbulence computations.
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